Use a complement for at least one and compare selection rules
Distinguish exactly one from at least one · Use no successes as the complementary event · Compare replacement and removal using complete paths
Section Paper 1 · Historical Core · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Complementary probability
- Two-stage trees
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Distinguish exactly one from at least one
- Use no successes as the complementary event
- Compare replacement and removal using complete paths
Translate the event first
For two red/blue selections, “exactly one red” includes RB and BR. “At least one red” also includes RR. Its complement is no red, meaning BB. Subtracting P(BB) from 1 can be simpler than adding three paths, but only after identifying the complete complementary event.
The selection rule changes the answer
From 3 red and 2 blue, two replaced draws give P(at least one red)=1−4/25=21/25. Two unreplaced draws give 1−1/10=9/10. The same starting contents do not imply the same result when the sampling process changes. Under either rule, add all four path probabilities to check the tree covers the sample space.
Sources for this lesson
Pause and explain
Does “at least one red” exclude RR?
Worked example
From 3 red and 2 blue, draw twice without replacement. Find P(at least one red).
Show the worked solution
- The only no-red path is BB, with probability (2/5)(1/4)=1/10.
- Subtract from 1.
Answer 9/10.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For two independent fair coin tosses, find P(at least one head).
Give me a hint
Use the complementary TT path.
Compare my reasoning
- TT has probability 1/4.
- The complement is 3/4.
3/4.
Look for these in your work
- Exclude only no-head outcomes
- Keep exactly one separate
From 1 red and 3 blue, draw twice with replacement. Find P(at least one red) and P(exactly one red).
Give me a hint
For the first use BB; for the second use RB and BR.
Compare my reasoning
- At least one=1−(3/4)²=7/16.
- Exactly one=2(1/4)(3/4)=6/16.
At least one 7/16; exactly one 3/8.
Look for these in your work
- Use different event definitions
- Include RR only where appropriate
Repeat those two events for 1 red and 3 blue without replacement. Explain why the two probabilities now agree.
Give me a hint
There cannot be two red in this experiment.
Compare my reasoning
- BB=(3/4)(2/3)=1/2, so at least one red=1/2.
- RB=(1/4)(3/3)=1/4 and BR=(3/4)(1/3)=1/4.
Both are 1/2, because RR is impossible without replacement.
Look for these in your work
- Update remaining counts
- Explain the zero RR probability
Common mistakes
- At least one includes two successes.
- The complement is no success, not exactly one failure.
What can you explain now?
Does “at least one red” exclude RR?
Compare with the explanation
No; two red still meets at least one.
The event includes one or more red outcomes.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does “at least one red” exclude RR?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.